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Testing for unit roots. II. (English) Zbl 0579.62014
Summary: [For part I see ibid. 49, 753-779 (1981; Zbl 0468.62021).]
This paper investigates the exact sampling distribution of the least squares estimator of \(\beta\) in the model \(y_ t=\mu +\beta y_{t- 1}+u_ t\) where the \(u_ t\) are independently \(N(0,\sigma^ 2)\). The distribution is calculated for the case where \(y_ 0\) is a known constant and where \(y_ 0\) is a random variable.
Given \(y_ 0\) is a constant we prove a small \(\sigma\) asymptotic result and compute the exact powers of nonsimilar tests of the random-walk hypothesis \(\beta =1\) and of the stability hypothesis \(\beta =0.9\). The exact powers of a test of the stability hypothesis are calculated for the case where \(y_ 0\) is random. The accuracy of the standard normal approximation is examined for both start-up regimes.

MSC:
62E15 Exact distribution theory in statistics
62P20 Applications of statistics to economics
62F03 Parametric hypothesis testing
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